Based on Lemma 2, we are able to derive the analytical BERs for the coded FTN signaling, where the ML detector considers finite ISI responses.
As previously described, \(\sigma _{{\text{R},\mathbf {e}}}^2\)  is derived by computing the expectation over all the possible transmitted sequences \(\mathbf {x}\) , while \(\sigma _{{\text{R}_L,\mathbf {e}}}^2\)  is determined based on the error sequence \(\mathbf {e}\) .
It is reasonable to employ \(\sigma _{{\text{R}_L,\mathbf {e}}}^2\)  to estimate the error probability for \(\mathbf {e}\) .
Moreover, to simplify the computation complexity, the distance spectrum of the FTN signaling can be searched based on the FTN error events [1]}, [2]}, [3]}.
For instance, given a CC error event \(\varepsilon \)  and a Hamming error weight \(w\) , let \(\Big {\lbrace }\big {(}D_{1_\varepsilon }, d^2(\mathbf {\tilde{e}}^{(w,1)}|\varepsilon ),d^2_{\text{ope}}(\mathbf {\tilde{e}}^{(w,1)}|\varepsilon )\big {)},...,\big {(}D_{M_\varepsilon },d^2(\mathbf {\tilde{e}}^{(w,M)}|\varepsilon ),,d^2_{\text{ope}}(\mathbf {\tilde{e}}^{(w,M)}|\varepsilon )\big {)}\Big {\rbrace }\)  be the distance spectrum for the CC error event \(\varepsilon \) , where \(D_{o_\varepsilon }\)  denotes the corresponding number of error sequences which have the same Euclidean and operative distance as \(\mathbf {\tilde{e}}^{(w,o)}\)  and \(M_\varepsilon \)  denotes the total number of considered error sequences.
According to Corollary 1, any two error sequences with the same Euclidean distance and operative Euclidean distance have the same values of \(\sigma _{\text{R}_L,\mathbf {e}}^2\) .
Therefore, the BER for a convolutionally encoded FTN system with a finite number of ISI responses can be approximated by:

Pb w = dminNcC'U(c, )oM 1A2K' INRNR/Kb-L+11Do
